The owners of the east London outfit, David Gold and David Sullivan, have zeroed in on the Israeli as the successor to Gianfranco Zola, according to The Mirror.
The Italian was sacked following a rocky season that saw the Hammers just about ensure survival, and the club then went on to issue a 10-point plan that listed the first priority as being the appointment of a new gaffer.
It has been reported that former Chelsea boss Grant will begin his duties on Tuesday, and will formally be unveiled as the manager of the club on Wednesday.
Croatia manager Slaven Bilic was also believed to be in the running, but Grant is said to have been the unanimous choice at Upton Park.
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